Optimal. Leaf size=98 \[ \frac {5}{21} a x^2 \sqrt [4]{a+b x^4}+\frac {1}{7} x^2 \left (a+b x^4\right )^{5/4}+\frac {5 a^{5/2} \left (1+\frac {b x^4}{a}\right )^{3/4} F\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{21 \sqrt {b} \left (a+b x^4\right )^{3/4}} \]
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Rubi [A]
time = 0.04, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {281, 201, 239,
237} \begin {gather*} \frac {5 a^{5/2} \left (\frac {b x^4}{a}+1\right )^{3/4} F\left (\left .\frac {1}{2} \text {ArcTan}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{21 \sqrt {b} \left (a+b x^4\right )^{3/4}}+\frac {5}{21} a x^2 \sqrt [4]{a+b x^4}+\frac {1}{7} x^2 \left (a+b x^4\right )^{5/4} \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 237
Rule 239
Rule 281
Rubi steps
\begin {align*} \int x \left (a+b x^4\right )^{5/4} \, dx &=\frac {1}{2} \text {Subst}\left (\int \left (a+b x^2\right )^{5/4} \, dx,x,x^2\right )\\ &=\frac {1}{7} x^2 \left (a+b x^4\right )^{5/4}+\frac {1}{14} (5 a) \text {Subst}\left (\int \sqrt [4]{a+b x^2} \, dx,x,x^2\right )\\ &=\frac {5}{21} a x^2 \sqrt [4]{a+b x^4}+\frac {1}{7} x^2 \left (a+b x^4\right )^{5/4}+\frac {1}{42} \left (5 a^2\right ) \text {Subst}\left (\int \frac {1}{\left (a+b x^2\right )^{3/4}} \, dx,x,x^2\right )\\ &=\frac {5}{21} a x^2 \sqrt [4]{a+b x^4}+\frac {1}{7} x^2 \left (a+b x^4\right )^{5/4}+\frac {\left (5 a^2 \left (1+\frac {b x^4}{a}\right )^{3/4}\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {b x^2}{a}\right )^{3/4}} \, dx,x,x^2\right )}{42 \left (a+b x^4\right )^{3/4}}\\ &=\frac {5}{21} a x^2 \sqrt [4]{a+b x^4}+\frac {1}{7} x^2 \left (a+b x^4\right )^{5/4}+\frac {5 a^{5/2} \left (1+\frac {b x^4}{a}\right )^{3/4} F\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{21 \sqrt {b} \left (a+b x^4\right )^{3/4}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 8.44, size = 52, normalized size = 0.53 \begin {gather*} \frac {a x^2 \sqrt [4]{a+b x^4} \, _2F_1\left (-\frac {5}{4},\frac {1}{2};\frac {3}{2};-\frac {b x^4}{a}\right )}{2 \sqrt [4]{1+\frac {b x^4}{a}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int x \left (b \,x^{4}+a \right )^{\frac {5}{4}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.08, size = 21, normalized size = 0.21 \begin {gather*} {\rm integral}\left ({\left (b x^{5} + a x\right )} {\left (b x^{4} + a\right )}^{\frac {1}{4}}, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.58, size = 29, normalized size = 0.30 \begin {gather*} \frac {a^{\frac {5}{4}} x^{2} {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, \frac {1}{2} \\ \frac {3}{2} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x\,{\left (b\,x^4+a\right )}^{5/4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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